The height, f, in metres, above the ground of a rider on a Ferris wheel can be modelled by the equation 𝑓(𝑡) = 5 sin (𝜋/3 𝑡) + 3, where t is the time, in seconds

a) What is the period of the function? [1 marks]

b) What are the maximum and minimum heights of the rider above the ground? [2 marks]

c) How high is the rider at 60 seconds? [2 marks]

d) What would the equation be if you needed to figure out the time for a rider to be at exactly 1 m?
[1 mark]

(a) sin(kt) has period 2π/k

(b) a sin(t) + k varies between k±a
(c) plug in t=60
(d) f(t) = 1

To answer these questions, let's analyze the given equation:

𝑓(𝑡) = 5 sin(𝜋/3 𝑡) + 3

a) The period of a sine function is given by T = (2𝜋)/b, where b is the coefficient of 𝑡 in the equation. In this case, b = (𝜋/3). So the period of the function is T = (2𝜋)/((𝜋/3)) = 6 seconds. Therefore, the period of the function is 6 seconds.

b) The maximum and minimum heights of the rider above the ground can be determined by looking at the amplitude of the sine function. The amplitude of a sine function is equal to the coefficient of the sine term. In this case, the coefficient is 5. So the maximum height above the ground is 5 + 3 = 8 meters, and the minimum height is -5 + 3 = -2 meters.

c) To find the height of the rider at 60 seconds, we substitute t = 60 into the equation:

𝑓(60) = 5 sin(𝜋/3 * 60) + 3
= 5 sin(20𝜋) + 3

Since sin(20𝜋) = 0 (since the sine function repeats every 2𝜋), the height at 60 seconds is:

𝑓(60) = 5 * 0 + 3
= 3 meters

Therefore, the rider is at a height of 3 meters at 60 seconds.

d) To figure out the time for a rider to be at exactly 1 meter, we set the equation equal to 1 and solve for t:

1 = 5 sin(𝜋/3 * t) + 3
-2 = 5 sin(𝜋/3 * t)
sin(𝜋/3 * t) = -2/5

To solve for t, we take the inverse sine of both sides:

𝜋/3 * t = arcsin(-2/5)
t = (3/arcsin(-2/5)) * 𝜋/3

So the equation to figure out the time for a rider to be at exactly 1 meter is:

𝑡 = (3/arcsin(-2/5)) * 𝜋/3

a) To find the period of the function, we need to determine the time it takes for a complete cycle of the sine function. The period (T) of any sine or cosine function is given by the formula T = 2π/|𝑏|, where 𝑏 is the coefficient of 𝑡 in the function.

In this case, the coefficient of 𝑡 is 𝜋/3. So, the period of the function is given by T = 2π/|𝜋/3|.

Calculating this gives T = 2π/(𝜋/3) = (6/𝜋) seconds.

Therefore, the period of the function is (6/𝜋) seconds.

b) The maximum and minimum heights of the rider above the ground occur at the maximum and minimum values of the sine function within one complete cycle. In the given equation, the coefficient of the sine function is 5, which represents the amplitude of the sine curve. The amplitude determines the maximum and minimum values.

The maximum height is given by Amplitude + Average. In this case, the amplitude is 5 and the average is 3. So, the maximum height is 5 + 3 = 8 meters.

The minimum height is given by Average - Amplitude. Therefore, the minimum height is 3 - 5 = -2 meters.

Thus, the maximum height of the rider above the ground is 8 meters, and the minimum height is -2 meters.

c) To find the height of the rider at 60 seconds, we substitute t = 60 into the equation 𝑓(𝑡) = 5 sin (𝜋/3 𝑡) + 3.

𝑓(60) = 5 sin (𝜋/3 * 60) + 3

Simplifying, we get 𝑓(60) = 5 sin (20𝜋) + 3.

Since the sine function oscillates between -1 and 1, we have 𝑓(60) = 5 * 1 + 3 = 8 meters.

Thus, the height of the rider at 60 seconds is 8 meters.

d) To find the equation to calculate the time for the rider to be at exactly 1 meter, we need to set 𝑓(𝑡) equal to 1 and solve for 𝑡.

1 = 5 sin (𝜋/3 𝑡) + 3

Subtracting 3 from both sides gives -2 = 5 sin (𝜋/3 𝑡).

Dividing by 5, we get -2/5 = sin (𝜋/3 𝑡).

Then, taking the inverse sine (or arcsine) of both sides, we have sin^(-1)(-2/5) = 𝜋/3 𝑡.

Now, we can solve for 𝑡:

𝑡 = (3/𝜋) * sin^(-1)(-2/5).

This is the equation to find the time for the rider to be at exactly 1 meter.

Note: Make sure to convert the result to seconds if necessary.